Blender editor scaling problem
Pivot Point

You set the 3D Cursor as transformation pivot point while the youtuber is using Median point (see the icon pointed by the mouse arrow in the orange box):



With the pivot point in the middle of selection he is able to scale the geometry from its center, not from side in your case. You can clearly see the pivot location by looking for the trasformation gizmo (colored arrows pointing towars axis)

You can change the pivot point in the button menu: